Security issues in electronic communications have been very important in the information age. Public key cryptographies (PKC) such as RSA and ECC (elliptic curve cryptosystems) have been adopted as key components for internet security, and in particular, for e-commerce systems authentication (electronic signatures) and secure communications. The RSA and ECC are mainly constructed from the complexity of integer factorization and discrete logarithm respectively. Although no proof is known for their NP-completeness or NP-hardness, both cryptosystems are still believed to be hard to break using convention systems. However, quantum computers have re-defined what problems are computational tractable and intractable, which has posed a new challenge to the security of classical cryptosystems.
The above-described background is merely intended to provide an overview of contextual information regarding networks, and is not intended to be exhaustive. Additional context may become apparent upon review of one or more of the various non-limiting embodiments of the following detailed description.